Carbon-12 in Nuclear Lattice EFT
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1 Carbon-12 in Nuclear Lattice EFT Nuclear Lattice EFT Collaboration Evgeny Epelbaum (Bochum) Hermann Krebs (Bochum) Timo A. Lähde (Jülich) Dean Lee (NC State) Thomas Luu (Jülich) Ulf-G. Meißner (Bonn/Jülich) Gautam Rupak (MS State) Bethe Forum on Methods in Lattice Field Theory Bonn, March 27, 2015, 10:15
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3 Outline Chiral EFT for nuclei on the lattice transfer matrix formalism Improved determination of phase shifts and low-energy constants Projection Monte Carlo and recent highlights for 12 C and 16 O 12 C as a life-generating element The Hoyle state and the Anthropic Principle
4 Lattice theory of pointlike protons and neutrons we do not resolve the quarks and gluons! a = 1.97 fm L = fm 8x8 lattice L = fm 6x6 lattice Work on smaller lattice spacings underway, a = fm also: larger volumes, antiperiodic boundary conditions Christopher Körber (Jülich)
5 Order-by-order expansion of the nucleon-nucleon potential: Epelbaum, Hammer, Meißner: Rev. Mod. Phys. 81, 1773 (2009) Currently implemented: 1) Improved LO - non-perturbative 2) NNLO - perturbative Extension to N 3 LO (also at smaller lattice spacings) Dechuan Du, Ning Li (FZ Jülich), José Alarcon (Bonn)
6 Transfer matrix OPE + 2 contact terms Transfer matrix + auxiliary fields + lattice Monte Carlo
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8 Need to determine the unknown coefficients > fix these from neutron-proton phase shifts! Asymptotically: Introduce a spherical wall : > get phase shifts from spectrum!
9 More accurate and systematical determination of 2NF constants at LO and NLO José Alarcon (Bonn), Ning Li (Jülich) Preliminary LO fits by José Alarcon (Bonn) Extension to higher orders underway
10 Next-to-leading order (NLO) contributions: more contact interactions Two-pion exchange potential (TPEP) > absorbed into contact terms at low cutoff Strong and electromagnetic isospin symmetry breaking
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13 Smaller lattice spacings forthcoming Fits (preliminary) to np scattering data by José Alarcon
14 Next-to-next-to-leading order (NNLO) three-nucleon force 3NF first appears at NNLO > contact, OPE and TPE contributions 2NF contribution at NNLO > again absorbed into contact terms at low cutoff
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16 Two physical couplings at LO > tuned to np scattering data
17 16 auxiliary field components 1 additional parameter from effective ranges
18 Pion propagator Instantaneous one-pion exchange
19 Use your favorite Monte Carlo algorithm, however: sign problem for the full transfer matrix The fermion determinant is small > relatively efficient algorithm
20 SU(4) symmetric transfer matrix M approx --> retain only C = C SU(4) non-zero! Still, need extrapolation in Euclidean time > should be independent of C SU(4)!
21 Allocations of supercomputing time (example for ) Resources provided by Forschungszentrum Jülich and RWTH Aachen (extension for under consideration :) - JUQUEEN (BG/Q, Jülich), 47 Mcore-h (project) + > 100 Mcore-h (institutional) - RWTH cluster (Intel, Aachen), 1.3 Mcore-h (project) Figure courtesy of Jülich Supercomputer Centre (JSC)
22 Projection Monte Carlo NLEFT calculations Lähde, Epelbaum, Krebs, Lee, Meißner, Rupak: Phys. Lett. B732, 110 (2014) Oxygen-16 ground state (LO) Neon-20 ground state (LO) Magnesium-24 ground state (LO) Silicon-28 ground state (LO)
23 NLO (NNLO) 2N force Isospin breaking NNLO 3N force Oxygen-16 ground state Neon-20 ground state
24 Lähde, Epelbaum, Krebs, Lee, Meißner, Rupak: Phys. Lett. B732, 110 (2014) a = 1.97 fm, L = 6 box errors statistical + extrapolation
25 Lähde, Epelbaum, Krebs, Lee, Meißner, Rupak: Phys. Lett. B732, 110 (2014) TPE potential absorbed into contact terms
26 Lähde, Epelbaum, Krebs, Lee, Meißner, Rupak: Phys. Lett. B732, 110 (2014) + contact 4N interaction, adjusted to A = 4 binding
27 Lähde, Epelbaum, Krebs, Lee, Meißner, Rupak: Phys. Lett. B732, 110 (2014) contact + nearest-neighbor 4N interaction (relevant for A > 12) Adjusted to A = 8 and A = 24
28 Requirements of 3NF: 1) Fit triton binding energy 2) Fit alpha particle binding energy 3) Stay numerically small!! Non-perturbative three-nucleon contact interaction? Alexander Rokash (Bochum)
29 Ground state of Carbon-12 Epelbaum, Krebs, Lähde, Lee, Meißner: Phys. Rev. Lett. 109, (2012)
30 Hoyle state Epelbaum, Krebs, Lähde, Lee, Meißner: Phys. Rev. Lett. 109, (2012)
31 NLEFT results for 12 C > Ground state and Hoyle State Epelbaum, Krebs, Lähde, Lee, Meißner: Phys. Rev. Lett. 109, (2012) Red > Plane wave trial wave functions Blue > Alpha cluster trial wave functions
32 NNLO NLEFT results for the low-energy spectrum of 12 C Epelbaum, Krebs, Lähde, Lee, Meißner: Phys. Rev. Lett. 109, (2012) Ground Hoyle Effects of rotational symmetry breaking on the lattice (Bing-Nan Lu, Jülich)
33 LO NLEFT results for charge radii and quadrupole moments in 12 C Epelbaum, Krebs, Lähde, Lee, Meißner: Phys. Rev. Lett. 109, (2012)
34 LO NLEFT results for transition matrix elements in 12 C Epelbaum, Krebs, Lähde, Lee, Meißner: Phys. Rev. Lett. 109, (2012) Extension to NNLO in the near future
35 Ground state of Oxygen-16 Epelbaum, Krebs, Lähde, Lee, Meißner, Rupak: Phys. Rev. Lett. 112, (2014)
36 First excited 0 + state of Oxygen-16 Epelbaum, Krebs, Lähde, Lee, Meißner, Rupak: Phys. Rev. Lett. 112, (2014)
37 Ground state of Oxygen-16 First excited 0 + state of Oxygen-16
38 LO NLEFT results for the EM properties of 16 O Epelbaum, Krebs, Lähde, Lee, Meißner, Rupak: Phys. Rev. Lett. 112, (2014) Rescaled > Approximate correction of overbinding effects at LO
39 Working around the sign problem in NLEFT Lähde, Luu, Lee, Meißner, Epelbaum, Krebs, Rupak, arxiv: first NLEFT calculation of 6 He full LO NLEFT Hamiltonian Wigner SU(4) symmetric Hamiltonian Perform calculations where the sign problem is more favorable -> Extrapolate to the physical point! Makes smaller lattice spacings and neutron-rich nuclei accessible
40 How is 12 C produced in red giant stars? The triple alpha process: Epelbaum, Krebs, Lähde, Lee, Meißner: PRL 110, (2013); EPJA 49, 82 (2010) What happens if we shift the Hoyle state slightly relative to the triple alpha threshold? Experiment: ± 0.18 kev
41 What happens to stellar nucleosynthesis if the Hoyle state is moved? Schlattl et al.: Astrophys. Space Sci. 291, 27 (2004) Very little 16 O produced: 12 C + 4 He -> 16 O too inefficient All 12 C converted to 16 O: 12 C + 4 He -> 16 O too efficient Anthropic bound on (ad hoc) variation of the Hoyle state We need to be able to relate this bound to shifts in the light quark mass!
42 Sources of quark mass (or pion mass) dependence in NLEFT Consider the leading-order NLEFT amplitude ChPT: Calculated to two loops (NNLO)
43 We compute the binding energies of all the states in NLEFT Sensitivity to small changes in the pion mass NLEFT Calculations ChPT, Lattice QCD Need to be re-expressed
44 Elimination of the (regularization-dependent) derivatives of the counterterms Differentiation of the Lüscher formula Two-nucleon problem in NLEFT ChPT, Lattice QCD
45 4 He NLEFT results for the pion-mass sensitivities Statistical + extrapolation error (parenthesis) + theoretical uncertainties 8 Be 12 C (ground) 12 C (Hoyle)
46 Epelbaum, Krebs, Lähde, Lee, Meißner: Phys. Rev. Lett. 110, (2013); EPJA 49, 82 (2013) Recall: Viability of carbon-oxygen based life:
47 The present theoretical understanding of the inverse scattering lengths Berengut et al.: Phys. Rev. D 87, (2013) Will likely be much improved by Lattice QCD in the near future!
48 The end of the world plot :) Epelbaum, Krebs, Lähde, Lee, Meißner: Phys. Rev. Lett. 110, (2013); EPJA 49, 82 (2013) Datapoint with errorbars > current knowledge of pion mass dependence of scattering lengths Berengut et al.: Phys. Rev. D 87, (2013) Lattice QCD may offer further insight in the near future
49 Thank you for your attention!
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